Is it possible to slide an edge beyond the two vertices its attached to?
Not as far as I know. If this is for an animation, have a look at booleans: https://www.blender.org/manual/modifiers/generate/booleans.html

They are implemented as a modifier, so you could have a hidden (hide it from the outliner by disabling it's render) circle (or rather a cylinder) acting as a "stencil" and then a plane that gradually grows in width. This plane would have the modifier set as "intersect" on it. This would make the plane only show in places that are inside the cylinder, thus creating an effect of the plane filling the circle.